Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system.

Explanation

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In physics and engineering, damping is mathematically modelled as a force with magnitude proportional to that of the velocity of the object but opposite in direction to it. Thus, for a simple mechanical damper, the force F is related to the velocity v by

$$

\bold{F} = -B \bold{v}

where B is the damper constant.

This relationship is perfectly analogous to electrical resistance. See Ohm's law.

In playing stringed instruments such as guitar or violin, damping is the quieting or abrupt silencing of the strings after they have been sounded, by pressing with the edge of the palm, or other parts of the hand such as the fingers on one or more strings near the bridge of the instrument. The strings themselves can be modelled as a continuum of infinitesimally small mass-spring-damper systems where the damping constant is much smaller than the resonant frequency, creating damped oscillations (see below). See also Vibrating string.

Example: mass-spring-damper

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An ideal mass-spring-damper system with mass m (in kilograms), spring constant k (in newtons per meter) and damper constant B (in newton-seconds per meter) can be described with the following formulae:

$$

F_\mathrm{s} \ \ = \ \ - k x

$$

F_\mathrm{d} \ \ = \ \ - B v \ \ = \ \ - B \dot{x} \ \ = \ \ - B \frac{dx}{dt}

Treating the mass as a free body and applying Law of Motion, we have:

$$

\Sigma\ F \ \ = \ \ ma\ \ = \ \ m \ddot{x} \ \ = \ \ m \frac{d^2x}{dt^2}

where a is the acceleration (in meters per second²) of the mass and $x$ is the displacement (in meters) of the mass relative to a fixed point of reference.

Differential equation

The equations of motion combine to form a second-order differential equation for displacement x as a function of time t (in seconds):

$m\; \backslash ddot\{x\}\; +\; B\; \backslash dot\{x\}\; +\; k\; x\; =\; 0$

Rearranging, we have

$\backslash ddot\{x\}\; +\; \{\; B\; \backslash over\; m\}\; \backslash dot\{x\}\; +\; \{k\; \backslash over\; m\}\; x\; =\; 0$

Next, to simplify the equation, we define the following parameters:

$\backslash omega\_0\; =\; \backslash sqrt\{\; k\; \backslash over\; m\; \}$

and

$\backslash zeta\; =\; \{\; B\; \backslash over\; 2m\; \}$

The first parameter, ω₀, is called the (undamped) natural frequency of the system. The second, ζ, is called the damping factor (not to be confused with loudspeaker damping factor). Both parameters represent angular frequencies and have for units of measure radians per second.

The differential equation now becomes:

$$

\ddot{x} + 2 \zeta \dot{x} + \omega_0^2 x = 0

Continuing, we can solve the equation by assuming

$\backslash \; x\; =\; e^\{\backslash gamma\; t\}\; \backslash$

where the parameter $\backslash \; \backslash gamma\; \backslash$ is, in general, a complex number.

Substituting this assumed solution back into the differential equation, we obtain:

$$

\gamma^2 + 2 \zeta \gamma + \omega_0^2 = 0

Solving for γ, we find:

$$

\gamma = - \zeta \pm \sqrt{\zeta^2 - \omega_0^2}

System behavior

The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω₀ and the damping factor ζ. Zeta, whose dimensions are those of frequency, can therefore be seen to be a quantity that that affects the damped resonant frequency of the system. When zeta is equal to the undamped frequency, the damped frequency is zero, and the system is critically damped. The ratio $\backslash sigma\; =\; \backslash zeta/\backslash omega\_0$ is known as the damping ratio, and describes the system's closeness to critical damping.

Critical damping

When $\backslash zeta^2\; -\; \backslash omega\_0^2\; =\; 0$ (or, equivalently, σ = 1), $\backslash gamma\; \backslash$ (defined above) is real and the system is critically damped. An example of critical damping is the door-closer seen on many hinged doors in public buildings.

Over-damping

When $\backslash zeta^2\; -\; \backslash omega\_0^2\; >\; 0$ (or σ > 1), $\backslash gamma\; \backslash$ is still real, but now the system is said to be over-damped. An overdamped door-closer will take longer to close the door than a critically damped door closer.

Under-damping

Finally, when (or σ < 1), $\backslash gamma\; \backslash$ is complex, and the system is under-damped. In this situation, the system will oscillate at the damped frequency, which is a function of the natural frequency and the damping factor.

The solution can be generally written as:

$$

x (t) \ = \ A e^{- \zeta t} cos( \omega_\mathrm{d} t + \phi)

where

$$

\omega_\mathrm{d} = \sqrt{\omega_0^2 - \zeta^2 }

represents the damped frequency of the system, and A and φ are determined by the initial conditions of the system (usually the initial position and velocity of the mass).

See also

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• Damping factor
• RLC circuit
• Oscillator
• Harmonic oscillator
• Simple harmonic motion
• Resonance
• Impedance
• Input impedance
• Output impedance
• Voltage divider
• Audio system measurements
• Bridging attenuation
• Matching attenuation

External links

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• Calculation of the matching attenuation,the damping factor, and the damping of bridging

Electronics terms | Control theory | Ordinary differential equations

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